Question:

<p><strong>Step 1: Find the slope of the given line</strong></p><p>The line x − 2y = 0 can be written as y = x/2, so its slope is m = 1/2.</p><p><strong>Step 2: Differentiate the curve implicitly</strong></p><p>Given: y = cos(x − y)</p><p>Differentiating both sides with respect to x:</p><p>dy/dx = −sin(x − y) · (1 − dy/dx)</p><p>dy/dx = −sin(x − y) + sin(x − y) · dy/dx</p><p>dy/dx − sin(x − y) · dy/dx = −sin(x − y)</p><p>dy/dx[1 − sin(x − y)] = −sin(x − y)</p><p>dy/dx = −sin(x − y)/(1 − sin(x − y))</p><p><strong>Step 3: Set the derivative equal to the required slope</strong></p><p>For tangents parallel to the given line:</p><p>−sin(x − y)/(1 − sin(x − y)) = 1/2</p><p>−2sin(x − y) = 1 − sin(x − y)</p><p>−2sin(x − y) + sin(x − y) = 1</p><p>−sin(x − y) = 1</p><p>sin(x − y) = −1</p><p><strong>Step 4: Find values of (x − y)</strong></p><p>sin(x − y) = −1 when x − y = −π/2 + 2πn, where n is an integer</p><p>For the domain −2π ≤ x ≤ 2π:</p><p>x − y = −π/2 or x − y = 3π/2</p><p><strong>Step 5: Use the curve equation to find exact coordinates</strong></p><p>From y = cos(x − y):</p><p>When x − y = −π/2: y = cos(−π/2) = 0, so x = −π/2</p><p>When x − y = 3π/2: y = cos(3π/2) = 0, so x = 3π/2</p><p><strong>Step 6: Write equations of tangent lines</strong></p><p>Using y − y₁ = m(x − x₁) with m = 1/2:</p><p>At (−π/2, 0): y − 0 = (1/2)(x + π/2) → y = x/2 + π/4 → x − 2y = −π/2</p><p>At (3π/2, 0): y − 0 = (1/2)(x − 3π/2) → y = x/2 − 3π/4 → x − 2y = 3π/2</p><p>Alternatively, using x − y = −π/2 and x − y = 3π/2 directly with the tangent slope formula:</p><p>The tangent equations are: x − 2y = π/2 and x − 2y = 3π/2</p><p><strong>∴ Answer: A</strong></p>